View source for Unimodular transformation

A [[Linear transformation|linear transformation]] of a finite-dimensional [[Vector space|vector space]] whose matrix has determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953801.png" />.



====Comments====
The name  "unimodular transformation"  is often restricted to mean a linear transformation with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953802.png" />. In the context of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953803.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953804.png" /> which is the quotient field of an integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953805.png" />, with a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953806.png" />-basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953807.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953808.png" />, a linear transformation is called unimodular if its matrix with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u0953809.png" /> has entries in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u09538010.png" /> and determinant a unit in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u09538011.png" />. Under each of these definitions the unimodular transformations form a group. In the case of linear transformations with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095380/u09538012.png" /> one often calls this the [[Unimodular group|unimodular group]], or, more commonly nowadays, the [[Special linear group|special linear group]].